Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = C24×AGL(1,𝔽5) (order 320) · Inn(G) = D5 · Out(G) = C25 |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2 |
| direct product | C3×C8×D5 |
| Fitting | C3×C5×C8 ⋊ C2 |
| derived, Sylow | C5 ⋊ C2×C3×C8 |
| Hall | C3×C5 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C8 | D5 |
| commutator, hypocenter | C5 | C2×C3×C8 |
| Frattini | C4 | C2×C3×D5 |
| Fitting | C3×C5×C8 | C2 |
| socle | C2×C3×C5 | C2×C4 |
| prime radical | C2×C3×D5 | C4 |
| cosocle | C4×C5 | C22×C3 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C3×C8 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,3)(4,5)(14,15,16);(1,2,4,5,3)(6,10,12,7,9,13,8,11)96 conjugacy classes — view the full character table.
| maximal subgroup of | D5×C3⋊C16, #90934, C3×D5×C16, … |
| maximal normal subgroup of | D5×C3⋊C16, #90934, C3×D5×C16, … |
| maximal quotient of | C3×D5×C16, C3×#3930, C3×C8×Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 33);